SAT-37: Graphing Parabolas — Intercepts, Vertex, and Symmetry
Graph any parabola fast by finding its intercepts, vertex, and axis of symmetry.
SAT-37: Graphing Parabolas — Intercepts, Vertex, and Symmetry
Description: A quadratic always graphs as a U-shaped curve called a parabola. If you can find a few special points — the intercepts and the vertex — you can sketch it perfectly. The SAT tests these points constantly.
The four things to find
- y-intercept: set x = 0. For y = ax2 + bx + c this is just c.
- x-intercepts (the roots): set y = 0 and solve the quadratic (factor or formula).
- Axis of symmetry: the vertical line x = −b/(2a).
- Vertex: sits on the axis of symmetry; its x is −b/(2a), then plug in for y.
(Oʻzbekcha: parabolani chizish uchun toʻrt narsa kerak: y-kesishma, x-kesishmalar (ildizlar), simmetriya oʻqi va uch.)
Why symmetry helps
A parabola is a mirror image across its axis of symmetry. So the two x-intercepts are always the same distance from the axis, and the vertex's x is exactly halfway between them.
Shortcut: if the x-intercepts are p and q, the axis (and vertex x) is the average (p + q) / 2. (Oʻzbekcha: ikkita x-kesishma oʻrtasi — simmetriya oʻqi.)
Worked Example
Graph y = x2 − 4x + 3.
- y-intercept: c = 3 → point (0, 3).
- x-intercepts: x2 − 4x + 3 = (x − 1)(x − 3) = 0 → x = 1 and x = 3 → points (1, 0) and (3, 0).
- Axis of symmetry: x = −(−4)/(2·1) = 2 (also the average of 1 and 3).
- Vertex: x = 2 → y = 22 − 4(2) + 3 = −1 → vertex (2, −1).
Since a = 1 > 0 the parabola opens up, so (2, −1) is the lowest point. (Oʻzbekcha: a > 0 boʻlgani uchun parabola yuqoriga ochiladi, uch eng past nuqta boʻladi.)
Practice
Find the vertex and axis of symmetry of y = x2 + 6x + 5.
Show answer
Axis: x = −6/(2·1) = −3. Vertex: y = (−3)2 + 6(−3) + 5 = 9 − 18 + 5 = −4 → (−3, −4).
(Check: x-intercepts are (x + 1)(x + 5) = 0 → x = −1 and −5, whose average is −3. ✓)
Key words — Kalit soʻzlar
- Parabola — parabola
- Intercept — kesishma nuqta
- x-intercept / Root — x-kesishma / ildiz
- y-intercept — y-kesishma
- Vertex — uch (choʻqqi)
- Axis of symmetry — simmetriya oʻqi
- Mirror image — koʻzgu aksi
- Opens up / down — yuqoriga / pastga ochiladi
- Average (midpoint) — oʻrtacha (oʻrta nuqta)
Summary
- y-intercept = c; x-intercepts come from solving y = 0.
- Axis of symmetry and vertex x are both −b/(2a).
- The vertex x is the average of the two x-intercepts.
- a > 0 opens up (vertex = minimum); a < 0 opens down (vertex = maximum).